A metro train traveling at a uniform speed of 90 km/h completely passes a station platform in 30 seconds. If the same train overtakes a track cyclist moving at 18 km/h in the same direction in 12 seconds, what is the length of the station platform (in meters)?

The correct answer is 510.

Step 1: Convert the speeds into meters per second (m/s).

Speed of the train = 90 km/h

To convert km/h to m/s, multiply by 5/18:

$\text{Speed of the train}=90\times \frac{5}{18}=25\text{m/s}$

Speed of the cyclist = 18 km/h

$\text{Speed of the cyclist}=18\times \frac{5}{18}=5\text{m/s}$

Step 2: Calculate the length of the train.

Since the train and the cyclist are moving in the same direction, the relative speed of the train with respect to the cyclist is the difference between their speeds:

$\text{Relative speed}=25-5=20\text{m/s}$

The train overtakes the cyclist in 12 seconds. The distance covered at this relative speed is equal to the length of the train:

$\text{Length of the train}=\text{Relative speed}\times \text{Time}$

$\text{Length of the train}=20\times 12=240\text{meters}$

Step 3: Calculate the length of the station platform.

Let the length of the station platform be L meters.

When the train completely passes the stationary platform in 30 seconds, the total distance covered equals the length of the train plus the length of the platform:

$\text{Total distance}=\text{Speed of the train}\times \text{Time}$

$240+L=25\times 30$

$240+L=750$

$L=750-240=510\text{meters}$

Therefore, the length of the station platform is 510 meters.